Uniqueness of the measurement function in Crofton's formula

This paper is dedicated to the memory of Wolfgang Weil, a kind colleague and teacher.
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Abstract

Crofton's intersection formula states that the (n−j)th intrinsic volume of a compact convex set in Rn can be obtained as an invariant integral of the (k−j)th intrinsic volume of sections with k-planes. This paper discusses the question if the (k−j)th intrinsic volume can be replaced by other functionals, that is, if the measurement function in Crofton's formula is unique.

The answer is negative: we show that the sums of the (k−j)th intrinsic volume and certain translation invariant continuous valuations of homogeneity degree k yield counterexamples. If the measurement function is local, these turn out to be the only examples when k=1 or when k=2 and we restrict considerations to even measurement functions. Additional examples of local functionals can be constructed when k≥2.

Section snippets

Uniqueness of local measurement functions in Crofton's formula

The classical Crofton formula [20] for compact convex sets K states that the invariantly integrated j-th intrinsic volume Vj of the intersection of K with a k-dimensional flat E is essentially an intrinsic volume of K:∫A(n,k)Vj(K∩E)μk(dE)=αn,j,kVn+j−k(K). Here μk is an (appropriately normalized) invariant measure on the space A(n,k) of all k-flats (k-dimensional affine subspaces of Rn), αn,j,k>0 is a known constant and 0≤j≤k≤n−1.

We will make use of the following notation. For a linear

Notation and preliminaries

Before giving the proofs of the above stated results we will introduce some further notation. Let A⊂Rn, we will denote its boundary by bd A, its interior by int A and its relative interior by relint A. The orthogonal complement of A is given by A⊥ and the convex hull by conv(A). If A is convex, its dimension is defined to be the dimension of its affine hull. The dual cone of A is given byA∘={x∈Rn:〈x,y〉≤0∀y∈A}. Note that the dual of the convex cone C={αa:a∈A and α≥0} of A satisfies C∘=A∘.

We have

Acknowledgments

The authors are supported by the Centre for Stochastic Geometry and Advanced Bioimaging, funded by the Villum Foundation.

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